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Chapter 7: Problem 1

What are the domain and range of \(\ln x ?\)

Short Answer

Expert verified
Answer: The domain of \(\ln x\) is \((0, \infty)\), and the range of \(\ln x\) is the set of all real numbers.

Step by step solution

01

Understand the natural logarithm function

The natural logarithm function, \(\ln x\), is the inverse function of \(y = e^x\), where \(e\) is Euler's number (\(\approx 2.71828\)). For a function to have an inverse, it must be both one-to-one and onto. Therefore, we will analyze the exponential function \(y = e^x\) first and then find the inverse of that function to determine the domain and range of \(\ln x\).
02

Analyze the exponential function \(y = e^x\)

The exponential function \(y = e^x\) is a continuous and strictly increasing function for all real numbers. This means that the domain of \(y = e^x\) is the set of all real numbers, and the range of \(y = e^x\) is the set of all positive real numbers, \((0, \infty)\) (since \(e^x\) is always positive for any real value of x).
03

Identify domain and range of \(\ln x\)

Since \(\ln x\) is the inverse function of \(y = e^x\), we can interchange the domain and range of the exponential function to find the domain and range for the natural logarithm function. The domain of \(e^x\) is the set of all real numbers, which is the range of \(\ln x\). The range of \(e^x\) is the set of all positive real numbers, \((0, \infty)\), which is the domain of \(\ln x\). Therefore, the domain of \(\ln x\) is \((0, \infty)\), and the range of \(\ln x\) is the set of all real numbers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inverse Function
An inverse function is a function that "undoes" the action of another function. If you have a function, say \( f(x) \), its inverse, denoted \( f^{-1}(x) \), will return the original input when applied to the output of \( f(x) \). In simpler terms, if you apply a function and then its inverse, you'll end up where you started.
Essentially, inverse functions reverse each other's processes.
  • For a function to have an inverse, it must be one-to-one; each input corresponds to exactly one unique output.
  • It must also be onto, meaning that every element of the range is mapped for some input.
Understanding inverse functions is crucial because they allow us to solve equations and understand relationships between quantities. In the case of the logarithm and exponential functions, they exhibit a special relationship where each is the inverse of the other.
Exponential Function
The exponential function, particularly the natural exponential function \( y = e^x \), is a fundamental mathematical function where the constant \( e \) (Euler's number) is raised to the power of \( x \). This function is critical in group of mathematical continuous functions, and is known for its unique properties.
  • It is always continuous and smooth, with a constant rate of growth.
  • It is defined for all real numbers, which means its domain is the set of all real numbers \(( -\infty, \infty)\).
  • Its range, on the other hand, only includes positive real numbers \((0, \infty)\), as \( e^x \) is always positive.
The exponential function grows exceptionally fast, and its inverse is the natural logarithm \( \ln x \). Understanding both the behavior and properties of \( y = e^x \) is essential in calculus, differential equations, and any domain involving exponential growth or decay.
Euler's Number
Euler's number, known as \( e \), is an irrational number approximately equal to 2.71828. It is named after the Swiss mathematician Leonhard Euler. This surprising yet important constant appears naturally in many areas of mathematics, especially in continuous growth and calculus.
Here are some key aspects of \( e \):
  • It is the base of the natural logarithm, which means \( \ln(e) = 1 \).
  • The number \( e \) can be defined as the sum of an infinite series, \( e = \sum_{n=0}^{\infty} \frac{1}{n!} \).
  • It serves as the base for natural exponential functions, which describe processes such as compound interest, population growth, and radioactive decay.
The properties of Euler's number make it invaluable in both theoretical and applied mathematics. Its presence in exponential and logarithmic functions highlight why understanding \( e \) is essential in the broader context of mathematical functions.

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Most popular questions from this chapter

Shallow-water velocity equation a. Confirm that the linear approximation to \(f(x)=\tanh x\) at \(a=0\) is \(L(x)=x\) b. Recall that the velocity of a surface wave on the ocean is \(v=\sqrt{\frac{g \lambda}{2 \pi} \tanh \left(\frac{2 \pi d}{\lambda}\right)} .\) In fluid dynamics, shallow water refers to water where the depth-to-wavelength ratio \(\frac{d}{\lambda}<0.05\) Use your answer to part (a) to explain why the approximate shallow-water velocity equation is \(v=\sqrt{g d}\) c. Use the shallow-water velocity equation to explain why waves tend to slow down as they approach the shore.

Chemotherapy In an experimental study at Dartmouth College, mice with tumors were treated with the chemotherapeutic drug Cisplatin. Before treatment, the tumors consisted entirely of clonogenic cells that divide rapidly, causing the tumors to double in size every 2.9 days. Immediately after treatment, \(99 \%\) of the cells in the tumor became quiescent cells which do not divide and lose \(50 \%\) of their volume every 5.7 days. For a particular mouse, assume the tumor size is \(0.5 \mathrm{cm}^{3}\) at the time of treatment. a. Find an exponential decay function \(V_{1}(t)\) that equals the total volume of the quiescent cells in the tumor \(t\) days after treatment. b. Find an exponential growth function \(V_{2}(t)\) that equals the total volume of the clonogenic cells in the tumor \(t\) days after treatment. c. Use parts (a) and (b) to find a function \(V(t)\) that equals the volume of the tumor \(t\) days after treatment. d. Plot a graph of \(V(t)\) for \(0 \leq t \leq 15 .\) What happens to the size of the tumor, assuming there are no follow-up treatments with Cisplatin? e. In cases where more than one chemotherapy treatment is required, it is often best to give a second treatment just before the tumor starts growing again. For the mice in this exercise. when should the second treatment be given?

Carbon dating The half-life of \(C-14\) is about 5730 yr. a. Archaeologists find a piece of cloth painted with organic dyes. Analysis of the dye in the cloth shows that only \(77 \%\) of the \(C\) -14 originally in the dye remains. When was the cloth painted? b. A well-preserved piece of wood found at an archaeological site has \(6.2 \%\) of the \(\mathrm{C}-14\) that it had when it was alive. Estimate when the wood was cut.

Points of intersection and area. a. Sketch the graphs of the functions \(f\) and \(g\) and find the \(x\) -coordinate of the points at which they intersect. b. Compute the area of the region described. \(f(x)=\operatorname{sech} x, g(x)=\tanh x ;\) the region bounded by the graphs of \(f, g,\) and the \(y\) -axis

Falling body When an object falling from rest encounters air resistance proportional to the square of its velocity, the distance it falls (in meters) after \(t\) seconds is given by \(d(t)=\frac{m}{k} \ln (\cosh (\sqrt{\frac{k g}{m}} t)),\) where \(m\) is the mass of the object in kilograms, \(g=9.8 \mathrm{m} / \mathrm{s}^{2}\) is the acceleration due to gravity, and \(k\) is a physical constant. a. A BASE jumper \((m=75 \mathrm{kg})\) leaps from a tall cliff and performs a ten-second delay (she free-falls for \(10 \mathrm{s}\) and then opens her chute). How far does she fall in 10 s? Assume \(k=0.2\) b. How long does it take for her to fall the first 100 m? The second \(100 \mathrm{m} ?\) What is her average velocity over each of these intervals?
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